Bertini theorems for Hilbert-Samuel multiplicity over finite fields

Abstract

Let $X$ be a quasiprojective subscheme of ${\mathbb{P}}_{{\mathbb{F}}_q}^n$. We prove a Bertini theorem for Hilbert--Samuel multiplicity of $X$; that is, there exists a positive-density set of hypersurfaces $H_f$ such that for every point $\xi \in X\cap H_f$, one has ${\mathrm{ord}}_{\xi }(f)=1$ and $e_{\xi }(X\cap H_f)=e_{\xi }(X)$. Furthermore, we extend this result on hypersurfaces to complete intersections, hypersurfaces containing a prescribed subscheme, and semiample linear systems.

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